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Has no other singularities close to it
complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number
Isolated_singularity
Location around which a function displays irregular behavior
essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essential singularity is a "left-over"
Essential_singularity
Point where a mathematical object behaves irregularly
(mathematics) Hyperbolic growth Movable singularity Pathological (mathematics) Regular singularity Singular solution "Singularities, Zeros, and Poles". mathfaculty
Singularity_(mathematics)
Undefined point on a holomorphic function which can be made regular
rigid that their isolated singularities can be completely classified. A holomorphic function's singularity is either not really a singularity at all, i.e.
Removable_singularity
Attribute of a mathematical function
The residue of a meromorphic function f {\displaystyle f} at an isolated singularity a {\displaystyle a} , often denoted Res ( f , a ) {\displaystyle
Residue_(complex_analysis)
Mathematical concept describing isolated singularity of an algebraic surface
a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex
Du_Val_singularity
Branch of mathematics studying functions of a complex variable
is applicable (see methods of contour integration). A "pole" (or isolated singularity) of a function is a point where the function's value becomes unbounded
Complex_analysis
}z^{n}={\frac {1}{1-z}}} with a singularity at z = 1 {\displaystyle z=1} . The example of the geometric series gives an isolated singularity. An example of a series
Vivanti–Pringsheim_theorem
Concept of complex analysis
it can be made to contain only the singularity of c {\displaystyle c} due to nature of isolated singularities. This may be used for calculation in
Residue_theorem
Concept in complex analysis
certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function
Zeros_and_poles
Mathematical theory
mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable
Singularity_theory
Theorem about the range of an analytic function
Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity at a point w {\textstyle w} , then on any punctured neighborhood of w
Picard_theorem
Phenomenon within general relativity
curvature singularity at the Cauchy horizon known as the mass-inflation singularity, the Cauchy horizon singularity, the infalling singularity, or the "fat
Mass_inflation
Class of mathematical function
{\displaystyle z=0} is an accumulation point of poles and is thus not an isolated singularity. The function f ( z ) = sin 1 z {\displaystyle f(z)=\sin {\frac
Meromorphic_function
Theorem in complex analysis
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Cauchy's_integral_theorem
Number of times a curve wraps around a point in the plane
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Winding_number
Awareness of facts
as explicit memory and can be learned through rote memorization of isolated, singular, facts. But in many cases, it is advantageous to foster a deeper understanding
Declarative_knowledge
Theorem about zeros of holomorphic functions
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Rouché's_theorem
Experimental operating system from Microsoft Research
Design Motivation and an overview of the Singularity Project Singularity source code on CodePlex Singularity: A research OS written in C# an interview
Singularity (operating system)
Singularity_(operating_system)
Mathematical function that preserves angles
often used to try to make models amenable to extension beyond curvature singularities, for example to permit description of the universe even before the Big
Conformal_map
Concept in algebraic geometry
it does not is given by the isolated singularity of x2 + y3z + z3 = 0 at the origin. Blowing it up gives the singularity x2 + y2z + yz3 = 0. It is not
Resolution_of_singularities
Statement in complex analysis
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Schwarz_lemma
Power series with negative powers
f(x)} for all x ∈ C {\displaystyle x\in \mathbb {C} } except at the singularity x = 0 {\displaystyle x=0} . The graph on the right shows f ( x ) {\displaystyle
Laurent_series
Type of function in mathematics
{\displaystyle 1} , because the nearest singularity is at z = − 1 {\displaystyle z=-1} . Complex singularities can determine the radius of convergence
Analytic_function
Provides integral formulas for all derivatives of a holomorphic function
. This is analytic (since the contour does not contain the other singularity). We can simplify f 1 {\displaystyle f_{1}} to be: f 1 ( z ) = z 2 z
Cauchy's_integral_formula
Second-order partial differential equation
converting interior problems into exterior problems, for studying isolated singularities, and for analyzing the behavior of harmonic functions at infinity
Laplace's_equation
Singularities of holomorphic functions extend infinitely outward
some direction. More precisely, it shows that an isolated singularity is always a removable singularity for any analytic function of n > 1 complex variables
Hartogs's_extension_theorem
(complex analysis) Residue (complex analysis) Isolated singularity Removable singularity Essential singularity Branch point Principal branch Weierstrass–Casorati
List of complex analysis topics
List_of_complex_analysis_topics
Method for assigning values to integrals
a singularity on an integral interval is avoided by limiting the integral interval to the non-singular domain. Depending on the type of singularity in
Cauchy_principal_value
Functions in mathematics
harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution
Harmonic_function
Theorem in complex analysis
h} is bounded and all the zeroes of g {\displaystyle g} are isolated, any singularities must be removable. Thus h {\displaystyle h} can be extended to
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Complex-differentiable (mathematical) function
a meromorphic function (defined to mean holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a
Holomorphic_function
Theorem in complex analysis
z-z_{Z}}+{g'(z) \over g(z)}.} Since g(zZ) ≠ 0, it follows that g' (z)/g(z) has no singularities at zZ, and thus is analytic at zZ, which implies that the residue of
Argument_principle
series (see Borel summation) and treats analytic functions with isolated singularities. He introduced the term in the late 1970s. Resurgent functions have
Resurgent_function
Theorem
center a {\displaystyle a} to the nearest non-removable singularity; if there are no singularities (i.e., if f {\displaystyle f} is an entire function),
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Integral criterion for holomorphy
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Morera's_theorem
Geometric representation of the complex numbers
convenience, because the points at which the infinite sum is undefined are isolated, and the cut plane can be replaced with a suitably punctured plane. In
Complex_plane
Characteristic property of holomorphic functions
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Cauchy–Riemann_equations
Study of systems of inequalitites
n {\displaystyle S^{n}} is the link of a real algebraic set with isolated singularity in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} 1981 Akbulut and King
Real_algebraic_geometry
related notion in algebraic geometry is the Milnor fiber of an isolated hypersurface singularity. This has a similar setup, where a polynomial f {\displaystyle
Milnor_map
Mathematical theorem
punctured neighborhood of z = 0. Hence it is an isolated singularity, as well as being an essential singularity. Using a change of variable to polar coordinates
Casorati–Weierstrass_theorem
Theorem on holomorphic functions
is non-constant and holomorphic. The roots of g {\displaystyle g} are isolated by the identity theorem, and by further decreasing the radius of the disk
Open mapping theorem (complex analysis)
Open_mapping_theorem_(complex_analysis)
Val singularities. Elliptic singularity (Kollár & Mori 1998, Theorem 5.22.) (Artin 1966) Artin, Michael (1966), "On isolated rational singularities of
Rational_singularity
Concept in complex analysis
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Antiderivative (complex analysis)
Antiderivative_(complex_analysis)
Mathematical theorem in complex analysis
|f(z)|} can only have a local minimum (which necessarily has value 0) at an isolated zero of f ( z ) {\displaystyle f(z)} . Another proof works by using Gauss's
Maximum_modulus_principle
Concept in Nielsen theory
{\displaystyle g(x)={\frac {x-f(x)}{||x-f(x)||}}.} Then g has an isolated singularity at x0, and maps the boundary of some deleted neighborhood of x0 to
Fixed-point_index
Invariant that plays a role in algebraic geometry and singularity theory
hypersurface singularity. Assume it is an isolated singularity: in the case of holomorphic mappings it is said that a hypersurface singularity f {\displaystyle
Milnor_number
Branch of mathematics
Nebojsa; Shinder, Evgeny (2021). "K-theory and the singularity category of quotient singularities". Annals of K-Theory. 6 (3): 381–424. arXiv:1809.10919
K-theory
Assignment of a vector to each point in a subset of Euclidean space
behaviour around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value −1 at a saddle singularity but +1 at a
Vector_field
Mathematical theorem
univalent. If the limit function is non-zero, then its zeros have to be isolated. Zeros with multiplicities can be counted by the winding number 1 2 π i
Riemann_mapping_theorem
Infinite sum that is considered independently from any notion of convergence
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Formal_power_series
Conformal mappings in complex analysis
regular singular points at z = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points
Schwarz_triangle_function
Russian-American mathematician (1963–2026)
1007/s00222-013-0453-4 with J. Kollár: Fundamental groups of links of isolated singularities. J. Amer. Math. Soc. 27 (2014), no. 4, 929–952. doi:10
Michael_Kapovich
Study of mathematical knots
{\displaystyle \mathbb {S} ^{n}} is the link of a real-algebraic set with isolated singularity in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} (Akbulut & King 1981)
Knot_theory
Computes the Poincaré–Hopf index of a real, analytic vector field at a singularity
Poincaré–Hopf index of a real, analytic vector field at an algebraically isolated singularity. It is named after David Eisenbud, Harold I. Levine, and George Khimshiashvili
Eisenbud–Levine–Khimshiashvili signature formula
Eisenbud–Levine–Khimshiashvili_signature_formula
French mathematician (born 1947)
functions, but these multi-valued functions have merely isolated singularities without singularities that form cuts with dimension one or greater. Écalle's
Jean_Écalle
General relativity model near spacetime singularities
relativity has a page on the topic of: BKL singularity A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe
BKL_singularity
Frobenius manifolds come from the singularity theory. Namely, the space of miniversal deformations of an isolated singularity has a Frobenius manifold structure
Frobenius_manifold
Type of mathematical curve
right is needed for having a true Weierstrass form. Singular cubics in Weierstrass form Isolated point y2 = x3 − x2 semicubical parabola y2 = x3 Double
Cubic_plane_curve
Theorem in complex analysis
Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Borel–Carathéodory_theorem
Association of one output to each input
analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the
Function_(mathematics)
Industrial robot
always singular in the sense that they can never span a six-dimensional twist space. This is often called an architectural singularity. A singularity is usually
Serial_manipulator
Finnish mathematician (born 1955)
thesis The Intersection Homology D-module on Hypersurfaces with Isolated Singularities. From 1983 to 1986 was a C. L. E. Moore instructor at the Massachusetts
Kari_Vilonen
Type of mathematical functions
Francesco Severi. Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function f : C n → C {\displaystyle
Function of several complex variables
Function_of_several_complex_variables
Knot which lies on the surface of a torus in 3-dimensional space
of an isolated complex hypersurface singularity. One intersects the complex hypersurface with a hypersphere, centred at the isolated singular point,
Torus_knot
Harmonic functions as solutions to Laplace's equation
behavior of isolated singularities of positive harmonic functions. As alluded to in the last section, one can classify the isolated singularities of harmonic
Potential_theory
On when a manifold that admits a singular foliation is homeomorphic to the sphere
the singularity a critical point of the function. The singularity is a center if it is a local extremum of the function; otherwise, the singularity is
Reeb_sphere_theorem
American mathematician (1942–2017)
analytically continued at least into the half-plane Re s > 0 except for an isolated singularity (presumably a simple pole) at s = 0." This should be "at s = 1" according
Paul_Chernoff
Vietnamese-French mathematician (1947–2025)
Terence Gaffney and David B. Massey). He was particularly concerned with singularity theory in the complex domain (Milnor fibrations, perverse sheaves). In
Lê_Dũng_Tráng
Infinite series that is not convergent
sometimes confused with zeta function regularization. If s = 0 is an isolated singularity, the sum is defined by the constant term of the Laurent series expansion
Divergent_series
French mathematician
contributions to algebraic geometry and commutative algebra, specifically to singularity theory, multiplicity theory and valuation theory. Teissier attained his
Bernard_Teissier
Mathematics study in geometry
This is related to the singularity category as follows: Given a superpotential W {\displaystyle W} with isolated singularities only at 0 {\displaystyle
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
Comic book series
new character named Singularity, a pocket universe that gains self-consciousness during "Secret Wars". Wilson likened Singularity to Q from Star Trek:
A-Force
Since, for holomorphic functions the sum of the residues at the isolated singularities plus the residue at infinity is zero, it can be expressed as: Res
Residue_at_infinity
Objects of certain abelian categories associated to topological spaces
which you move between singular CY target spaces require moving through either a small resolution or deformation of the singularity (T. Hubsch, 1992) and
Perverse_sheaf
American mathematician (born 1931)
dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on the topology of isolated singular points of complex
John_Milnor
British mathematician (1903–1987)
differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named after him. Du Val was born in Cheadle
Patrick_du_Val
Indian university teacher (born 1971)
concrete problems. His results on 0-cycles on algebraic varieties with isolated singularities effectively reduces their study to the corresponding study on the
Amalendu_Krishna
Point on a curve not given by a smooth embedding of a parameter
singular point at the origin. However, a node such as that of y 2 − x 3 − x 2 = 0 {\displaystyle y^{2}-x^{3}-x^{2}=0} at the origin is a singularity of
Singular_point_of_a_curve
Singularities of algebraic varieties
(1985) and Reid. In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Canonical_singularity
Cubic plane curve
the line at infinity is crossed by the asymptotic line. It has an isolated singular point, at the point where the line at infinity is crossed by its axis
Witch_of_Agnesi
{Z} /2\mathbb {Z} )} which is a singular subvariety of A 3 {\displaystyle \mathbb {A} ^{3}} with isolated singularity at ( 0 , 0 , 0 ) {\displaystyle
GIT_quotient
Compact astronomical body
hole would create a so-called naked singularity, a singularity outside of a black hole. Because these singularities make the universe inherently unpredictable
Black_hole
British mathematician
been in singularity theory as developed by R. Thom, J. Milnor and V. Arnold, and especially concerns the classification of isolated singularities of differentiable
C._T._C._Wall
Differential form in commutative algebra
Other counterexamples can be found in algebraic plane curves with isolated singularities whose Milnor and Tjurina numbers are non-equal. A proof of Grothendieck's
Kähler_differential
French mathematician (1789–1857)
of a function. This concept concerns functions that have poles—isolated singularities, i.e., points where a function goes to positive or negative infinity
Augustin-Louis_Cauchy
Term in mathematics
Scientific, p. 380, ISBN 978-981-02-0662-8 Looijenga, E. J. N. (1984), Isolated singular points on complete intersections, London Mathematical Society Lecture
Complete_intersection
i {\displaystyle f_{i}} are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of f {\displaystyle f} is isomorphic
Thom–Sebastiani_theorem
3 {\displaystyle \mathbb {V} (f)\subset \mathbb {C} ^{3}} has an isolated singularity at the origin since f ( 0 ) = 0 {\displaystyle f(0)=0} and all partial
Intersection_homology
Concept in algebraic geometry
scheme of dimension 2 has only isolated singularities. Normalization is not usually used for resolution of singularities for schemes of higher dimension
Normal_scheme
Latvian-American mathematician (1914–1993)
proved an extension of Riemann's theorem on removable singularities, showing that any isolated singularity of a pencil of minimal surfaces can be removed; he
Lipman_Bers
Japanese mathematician (born 1944)
Quasihomogene isolierte Singularitäten von Hyperflächen (Quasihomogeneous isolated singularities of hypersurfaces). Saito is a professor at the Research Institute
Kyoji_Saito
American mathematician
Stern, Pure hodge structure on the L2-cohomology of varieties with isolated singularities, Journal fur die Reine und Angewandte Mathematik, vol. 533 (2001)
Mark_Stern
Exact solution for the Einstein field equations
outside that horizon. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different coordinate system.
Kerr_metric
Region in spacetime from which nothing can escape
fundamental gravitational collapse models, an event horizon forms before the singularity of a black hole. If all the stars in the Milky Way would gradually aggregate
Event_horizon
American mathematician
"Refined asymptotics for constant scalar curvature metrics with isolated singularities". Inventiones Mathematicae. 135 (2): 233–272. arXiv:math/9807038
Rafe_Mazzeo
Technique for wavelet analysis
effect that increases the number of large coefficients produced by isolated singularities. Each lifting step maintains the filter biorthogonality but provides
Lifting_scheme
Sino-Tibetan of India
Kanashi is a Sino-Tibetan language spoken in the isolated Malana (Malani) village area in Kullu District, Himachal Pradesh, India. It is, to some extent
Kanashi_language
Isolated point in the solution set of a polynomial equation in two real variables
in two complex variables can never have an isolated point. An acnode is a critical point, or singularity, of the defining polynomial function, in the
Acnode
is analogous to the following problem: Is the local topology at an isolated singular point of a minimal hypersurface necessarily different from that of
Spherical_Bernstein's_problem
ISOLATED SINGULARITY
ISOLATED SINGULARITY
Surname or Lastname
English (Devon)
English (Devon) : from Middle English hauek ‘hawk’, applied as a metonymic occupational name for a hawker (see Hawker), a name denoting a tenant who held land in return for providing hawks for his lord, or a nickname for someone supposedly resembling a hawk. There was an Old English personal name (originally a byname) H(e)afoc ‘hawk’, which persisted into the early Middle English period as a personal name and may therefore also be a source.English (Devon) : topographic name for someone who lived in an isolated nook, from Middle English halke (derived from Old English halh + the diminutive suffix -oc), or a habitational name from some minor place named with this word, such as Halke in Sheldwich, Kent.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : from a Germanic personal name composed of the elements folk ‘people’ + hari, heri ‘army’, which was introduced into England from France by the Normans; isolated examples may derive from the cognate Old English Folchere or Old Norse Folkar, but these names were far less common.
Surname or Lastname
English
English : habitational name from any of the various places so called. The majority, with examples in at least fourteen counties, get the name from Old English hÅh ‘ridge’, ‘spur’ (literally ‘heel’) + tÅ«n ‘enclosure’, ‘settlement’. Haughton in Nottinghamshire also has this origin, and may have contributed to the surname. A smaller group of Houghtons, with examples in Lancashire and South Yorkshire, have as their first element Old English halh ‘nook’, ‘recess’. In the case of isolated examples in Devon and East Yorkshire, the first elements appear to be unattested Old English personal names or bynames, of which the forms approximate to Huhha and Hofa respectively, but the meanings are unknown.
Girl/Female
Arabic, Muslim, Sindhi
Singularity
Girl/Female
Muslim/Islamic
Singularity
Boy/Male
Muslim
Singularity
Surname or Lastname
English and Scottish
English and Scottish : habitational name from any of the numerous places so called. The vast majority, including those in Cambridgeshire, Cumbria, Dumfries, County Durham, Kent, Lancashire, Lincolnshire, Norfolk, Northumberland, Oxfordshire, Sussex, and West Yorkshire, are named from Old English denu ‘valley’ (see Dean 1) + tūn ‘enclosure’, ‘settlement’. An isolated example in Northamptonshire appears in Domesday Book as Dodintone ‘settlement associated with Dodda’.
Boy/Male
Indian
Singularity
Surname or Lastname
English
English : habitational name from any of several places called Dockray, of which there are four examples in Cumbria. A possible origin of the place name is Old Norse d{o,}kk ‘hollow’, ‘valley’ + vrá ‘isolated place’; the first element is, however, more likely to be Old English docce ‘dock’ (the plant).Irish : reduced Anglicized form of Gaelic Ó Dochraidh ‘descendant of Dochradh’, a personal name that is a variant of Dochartach (see Doherty).
ISOLATED SINGULARITY
ISOLATED SINGULARITY
Surname or Lastname
Korean
Korean : variant of Paek.English : variant of Pack.
Boy/Male
Indian, Modern
God; Dianne
Boy/Male
Hindu
Principled
Boy/Male
Tamil
Lord Murugan
Boy/Male
Muslim
Supported
Boy/Male
Japanese
Spread light.
Surname or Lastname
English
English : habitational name from places called Swinford in Oxfordshire and Leicestershire, from Kingswinford in Staffordshire, or from Old Swinford in Worcestershire, named with Old English swīn ‘swine’, ‘hog’ + ford ‘ford’.
Boy/Male
Bengali, Hebrew, Hindu, Indian, Sanskrit
Loved
Girl/Female
Hindu
Test, Exam
Girl/Female
Indian, Kannada
Gold
ISOLATED SINGULARITY
ISOLATED SINGULARITY
ISOLATED SINGULARITY
ISOLATED SINGULARITY
ISOLATED SINGULARITY
v. t.
To insulate. See Insulate.
v. t.
To separate from all foreign substances; to make pure; to obtain in a free state.
a.
Distended or enlarged fictitiously; as, inflated prices, etc.
v. t.
To place in a detached situation; to place by itself or alone; to insulate; to separate from others.
p. pr. & vb. n.
of Insolate
a.
Inflated; bombastic.
imp. & p. p.
of Isolate
p. a.
Standing by itself; not being contiguous to other bodies; separated; unconnected; isolated; as, an insulated house or column.
p. pr. & vb. n.
of Isolate
a.
Inflated with wind.
a.
Flushed, inflated.
p. a.
Blown in; inflated.
n.
One who, or that which, isolates.
n.
An idolater.
a.
Turgid; swelling; puffed up; bombastic; pompous; as, an inflated style.
a.
Inflated; boastful.
imp. & p. p.
of Insolate
a.
Placed or standing alone; detached; separated from others.
a.
Filled, as with air or gas; blown up; distended; as, a balloon inflated with gas.
adv.
In an isolated manner.